The Biquaternions

The Biquaternions

Renee RussellKim KestingCaitlin HultSPWM 2011

Sir William Rowan Hamilton(1805-1865)

Physicist, Astronomer and Mathematician

“This young man, I do not say will be, but is, the first

mathematician of his age” – Bishop Dr. John Brinkley

• Optics• Classical and Quantum Mechanics • Electromagnetism

• Algebra:• Discovered Quaternions & Biquaternions!

Contributions to Science and Mathematics:

Review of Quaternions, H

A quaternion is a number of the form of:

Q = a + bi + cj + dk

where a, b, c, d R, and i2 = j2 = k2 = ijk = -1.

So… what is a biquaternion?

Biquaternions• A biquaternion is a number of the form

B = a + bi + cj + dk

where ,

and i2 = j2 = k2 = ijk = -1.

a, b, c, d C

CONFUSING:

(a+bi) + (c+di)i + (w+xi)j + (y+zi)k

Biquaternions

We can avoid this confusion by renaming i, j,and k:

B = (a +bi) + (c+di)e1 +(w+xi)e2 +(y+zi)e3

e12 = e2

2 = e32 =e1e2e3 = -1.

* Notice this i is different from the i component of the basis, {1, i, j, k} for a (bi)quaternion! *

B can also be written as the complex combination of two quaternions:

B = Q + iQ’ where i =√-1, and Q,Q’ H.

B = (a+bi) + (c+di)e1 + (w+xi)e2 + (y+zi)e3

=(a + ce1 + we2 +ye3) +i(b + de3 + xe2 +ze3)

where a, b, c, d, w, x, y, z R

Biquaternions

Properties of the BiquarternionsADDITION:

• We define addition component-wise:

B = a + be1 + ce2 + de3 where a, b, c, d C B’ = w + xe1 + ye2 + ze3 where w, x, y, z C

B +B’ =(a+w) + (b+x)e1 +(c+y)e2 +(d+z)e3

Properties of the Biquarternions

Properties of the Biquarternions

Oh yeah!

Properties of the BiquarternionsMULTIPLICATION:

• The formula for the product of two biquaternions is the same as for quaternions:

(a,b)(c,d) = (ac-db*, a*d+cb) where a, b, c, d C.

•Closed•Associative•NOT Commutative•Identity:

1 = (1+0i) + 0e1 + 0e2 + 0e3

Biquaternions are an algebra

over C! biquaterions

Properties of the Biquarternions

So far, the biquaterions over C have all the same properties as the quaternions over R.

DIVISION?In other words, does every non-zero element have a

multiplicative inverse?

Properties of the BiquarternionsRecall for a quaternion, Q H,

Q-1 = a – be1 – ce2 – de3 where a, b, c, d R a2 + b2 + c2 + d2

Does this work for biquaternions?

Biquaternions are NOT a division algebra over C!

Quaternions(over R)

Biquaternions

(over C)Vector Space? ✔ ✔Algebra? ✔ ✔Division Algebra?

✔ ✖Normed Division Algebra?

✔ ✖

Biquaternions are isomorphic to M2x2(C)

Define a map f: BQ M2x2(C) by the following:

f(w + xe1 + ye2 + ze2 ) = w+xi y+zi -y+zi w-xi

where w, x, y, z C.

We can show that f is one-to-one, onto, and is a linear transformation. Therefore, BQ is isomorphic to M2x2(C).

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Applications of Biquarternions

• Special Relativity• Physics• Linear Algebra• Electromagnetism